A $20-20-$ resistor, $50-$F$50-$F capacitor, and $30-$mH inductor are connected in series with an ac source of amplitude $10$ V and frequency $125$ Hz$.($a$)$ What is the impedance of the circuit? $($b$)$ What is the amplitude of the current in the circuit? $($c$)$ What is the phase constant of the current? Is it leading or lagging the source voltage? $($d$)$ Write voltage drops across the resistor, capacitor, and inductor and the source voltage as a function of time. $($e$)$ What is the power factor of the circuit? $($f$)$ How much energy is used by the resistor in $2\mathrm{.}5$ s$?$

Suppose a coil has a self$-$inductance of $20\mathrm{.}0$ H and a resistance of $200200.$ What $($a$)$ capacitance and $($b$)$ resistance must be connected in series with the coil to produce a circuit that has a resonant frequency of $100$ Hz and a Q of $10?$